Long-Run Risk


Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University


A common belief is that the law of averages will rule in the long run, eliminating risk.


It’s true that the average outcome from a gamble should not be risky in the long run.


But the gain or loss from gambling is average gain per gamble \(\times\) number of gambles.

Betting on the stock market

  • Based on history, the bet is definitely in our favor. We have much better odds than in a casino.

  • If we play for a long time, we will almost certainly come out ahead.

  • But how far ahead is quite uncertain.

  • Let’s look at the best and worst periods in the past.

Simulate returns

  • Mean and std dev of U.S. market return 1970-2021 was 12.5% and 17.4%.
  • Simulate 30-year compounded returns.
import numpy as np

mn = 0.125
sd = 0.174

r = np.random.normal(loc=mn, scale=sd, size=30)
comp_ret = np.prod(1+r)

Repeat and evaluate the distribution

nsims = 1000

r = np.random.normal(loc=mn, scale=sd, size=30*nsims)
r = r.reshape((30,1000))
comp_ret = np.prod(1+r, axis=0)

Simulating a Single History

Note the variation in compounded returns in different simulations

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Simulating Multiple Histories

  • The variation in outcomes is captured by the standard dev.
  • Compounding \(\Rightarrow\) positive skewness
  • Mean outcome \(>\) Median outcome
  • Lower risk (for a given mean) \(\Rightarrow\) higher median

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Retirement Planning Simulation

Same exercise as in our first module

But generate a random return each month of the investment lifetime

And simulate many lifetimes in this way

BBCX Investment Library

HTML tutorial

If we want to forecast a constant return (not simulate) and base the return on history, should we use the historical arithmetic average or historical geometric average?

Geometric average is better for forecasting future compounded returns.

Arithmetic average is better for forecasting future (arithmetic) average return.

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